double-counting poker combinations

I'm trying to calculate the hand odds of 5-card stud with 4 and 5-card poker hands. To clarify all the normal poker hands with the addition of 4-card flushes and straights. For calculation purposes, a 4-card flush outranks a 4-card straight. Each of those hands lose to a three of a kind and beat a two pair. A math check is in dire need.

The question comes with calculating combos with a pair/ high card. With a pair in 5-card poker you would just calculate the odds of a pair with 3 singletons. With the possibility of 4-card hands, those 3 singletons could complete the poker hands. Here's what I have so far:



right now in my sheet I'm double-counting ~7k hand combinations and can't figure out where they're coming from. numbers where "I calculated" could easily be wrong. Thanks for any help

Quote: richodude

I'm trying to calculate the hand odds of 5-card stud with 4 and 5-card poker hands. To clarify all the normal poker hands with the addition of 4-card flushes and straights. For calculation purposes, a 4-card flush outranks a 4-card straight. Each of those hands lose to a three of a kind and beat a two pair. A math check is in dire need.

The question comes with calculating combos with a pair/ high card. With a pair in 5-card poker you would just calculate the odds of a pair with 3 singletons. With the possibility of 4-card hands, those 3 singletons could complete the poker hands. Here's what I have so far:

pair=13*COMBIN(4,2)*(COMBIN(12,3)-(44/13))*(4^3-2)+44*16

The explanation is 13 possible ranks for the pair picking out 2 of 4 suits for that pair. we pick 3 more ranks out of 12 remaining for the singletons (all standard so far) and I subtract the 44 stretches (2 given pair of 2's, 4 given a pair of 7's, etc.) over the 13 ranks. I then multiply by 4^3 for the possible suits of the singletons and subtract 2 for the cases where all 3 singletons match either paired card. I then add back the straight flush with a pair possibilities because I think I double-subtracted them?

high card=(COMBIN(13,5)-10-78)*(4^5-24)

choosing 5 singletons out of 13 ranks and subtracting the 10 5straight stretches and 78(I calculated) 4straight stretches multiplied by 4^5 for all the suits of the singletons minus 4 5flushes and 20 4flushes

4flush=4*(COMBIN(13,4)-11)*39-120

4 suits times picking 4 cards out of 13 ranks - 11 straight flush stretches * 39 non-suited kickers(5 card flush would outrank) - 120 5-card straights with 4 suited cards (I calculated)

4straight=9*(4^4-4)*40+2*(4^4-4)*44-176*10

9 4straights with 2 kicker ranks that would complete a 5straight (ie. not A,2,3,4 or J,Q,K,A) times 4 suits for the 4 cards - 4 sflushes times 40 kickers that wont improve the hand added to that 2 4straights of the same thing times 44 kickers that wont improve the hand. I then subtract the 4straight hands that are also 4flush hands (again that I calculated) since flush outranks straight

right now in my sheet I'm double-counting ~7k hand combinations and can't figure out where they're coming from. numbers where "I calculated" could easily be wrong. Thanks for any help
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Are you subtracting out the lower hands if they are also part of a higher qualifying hand?

For example, you could have both a pair and a 4 card straight or a pair and a 4 card flush. You don't want to count both of them.

Is this just a curiosity or is the correct answer important to you? The way my VP program is structured, I could get the flop counts fairly quickly.

Start with 5 card royal & straight flushes, then 4 card royal & straight flushes and work your way down from there to a pair and hi card hands. This makes it easier to subtract 5 card straight flushes from 4 card straight flushes, and to subtract straight flushes from flushes and straights.

Make separate categories out of 4 card straight plus a pair, 4 card flush plus a pair, and 4 card SF plus a pair.

I did a quick bit of coding for these hands. Maybe you can check to see if the results are correct. I don't want to spend much time debugging this myself, but if you can point out obvious problems I will try to correct my code.

I count 12,288 low straights A234, 135,168 straights not using the Ace as a low card, and 55,770 high flushes. Now, removing all hands that also have flushes, straights, or SFs, I get:

EDIT: I think I am double counting the low 4-straights. The totals I gave above have 11 times more straights than low straights. The straights can start with 11 ranks, so 135,168 includes all 4-straights.

I found a feature was turned off in my program regarding flushes assumed to be higher than straights. In this case, the ordinary straight is higher than the 4-flush. Turning on the reordering code, I get this result:

My diagnostic hand listings use colors for suits on my xterm console. I should figure out how to list all the hands in color using the formatting codes here. Then I could post a list of all of the hands here.

Quote: gordonm888

Make separate categories out of 4 card straight plus a pair, 4 card flush plus a pair, and 4 card SF plus a pair.
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This bit is really helpful. I started again today with a fresh mind and think I've got it. The cells marked "(calculated)" are my answers

There is a special case where a 4-card straight with a pair where the paired card also forms a 4-card flush. It happens to be that every single scenario of this forms a 4 card straight flush which I already subtracted out.
These calculations help me with the whole hierarchy

I'm somewhat confident in these numbers but now have some decisions about the game. Should 4-card flush still outrank straight and two pair? The difference is little and would be more intuitive for people without the numbers and knowledge of 5-card hands. Next, how would I add sections for ace high/king high/queen high etc. combinations? Normally it wouldn't be so bad but the 4-card combos really screw you.

Commenting on Mental's calculations, I'm curious as to how your code got lower answers than me. The 4flush is definitely too low at only 3x more likely than a 5flush but the 4straight is ballpark. I like the video poker prediction but this will be a stud game of player vs. dealer. Hoping to get into game invention and looking at a class in LV in August where corporations can buy final projects (table game ideas).

Quote: richodude

Commenting on Mental's calculations, I'm curious as to how your code got lower answers than me. The 4flush is definitely too low at only 3x more likely than a 5flush but the 4straight is ballpark. I like the video poker prediction but this will be a stud game of player vs. dealer. Hoping to get into game invention and looking at a class in LV in August where corporations can buy final projects (table game ideas).
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To be clear, I am not calculating the number of 4-card hands. I am calculating the number of 4-card hands that are not contained within a higher paying hand. An 5-card flush contains 5 different 4-card flushes, right? A 5-card straight contains two different 4-card straights.

I am calculating the 5-card stud probabilities for getting paid for a certain hand, because that is what you need for the EV. I did not look at your spoiler, but I suppose you are just getting the raw numbers without accounting for counterfeiting.

If I make the 4-straight pay more than the 5-flush and 5-straight I will get a larger number of 4-straights. I will look to see if I am catching al 4-flushes tomorrow.

Quote: Mental

To be clear, I am not calculating the number of 4-card hands. I am calculating the number of 4-card hands that are not contained within a higher paying hand. An 5-card flush contains 5 different 4-card flushes, right? A 5-card straight contains two different 4-card straights.

I am calculating the 5-card stud probabilities for getting paid for a certain hand, because that is what you need for the EV. I did not look at your spoiler, but I suppose you are just getting the raw numbers without accounting for counterfeiting.

If I make the 4-straight pay more than the 5-flush and 5-straight I will get a larger number of 4-straights.
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Sorry if any of my words seem pointless/rude, I'm just trying to be thorough

"A 5-card flush contains 5 different 4-card flushes, right?" Yes. The hand A♥2♥3♥4♥5♥ contains 1 5-card [straight] flush and 5 4-card flushes (one for every rank removed). However there should be considerably more due to the possibility of hands like A♥2♥3♥4♥5♣. Ignoring the 5straight and 4sflush, the 5th card is not a heart meaning an extra 39 kickers for a lot of extra combos. but there are also many 4flush combos where the kicker forms a 5straight which outranks and needs to be subtracted.

"A 5-card straight contains two different 4-card straights." Also yes, but a 5straight outranks a 4straight so shouldn't be counted. Your numbers seem to justify your process though, but our results are different somehow.

"If I make the 4-straight pay more than the 5-flush and 5-straight I will get a larger number of 4-straights." Yes but I don't think that difference is why our results differ. Our 4straight combos differ by almost 30k, but the 4sflush, 4flush w/ 4straight, and 5straight/sflush hands (the hands containing 4straights that have something better) don't even total 15k (according to my results) That means one or both of our calcs are wrong.

If you think any of my logic is wrong, please let me know

Okay, I just looked at my code and saw a single character typo in the 4-flush code.
I was definitely undercounting 4-flushes because my loop did not start at 0. We are getting closer, but I am too drunk to look at it anymore tonight.